Average Error: 41.5 → 0.7
Time: 3.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\
\;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\end{array}
double f(double x) {
        double r119096 = x;
        double r119097 = exp(r119096);
        double r119098 = 1.0;
        double r119099 = r119097 - r119098;
        double r119100 = r119097 / r119099;
        return r119100;
}
↓
double f(double x) {
        double r119101 = x;
        double r119102 = exp(r119101);
        double r119103 = 0.8526603959516881;
        bool r119104 = r119102 <= r119103;
        double r119105 = 1.0;
        double r119106 = -r119105;
        double r119107 = r119101 + r119101;
        double r119108 = exp(r119107);
        double r119109 = fma(r119106, r119105, r119108);
        double r119110 = r119102 / r119109;
        double r119111 = r119102 + r119105;
        double r119112 = r119110 * r119111;
        double r119113 = 0.08333333333333333;
        double r119114 = 1.0;
        double r119115 = r119114 / r119101;
        double r119116 = fma(r119113, r119101, r119115);
        double r119117 = 0.5;
        double r119118 = r119116 + r119117;
        double r119119 = r119104 ? r119112 : r119118;
        return r119119;
}

Error

Bits error versus x

Target

Original41.5
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.8526603959516881

    1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied flip--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]

    4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)\]

    if 0.8526603959516881 < (exp x)

    1. Initial program 62.0

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

    3. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))